Lesson 21: Decimals: Introduction and Addition/Subtraction

§Text Passage

1Summary 重點摘要

Decimal addition/subtraction: line up the decimal point. e.g. 3.25 + 1.7 = 4.95. Learn place value and operations.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Block 1: Decimal Place ValueThe place values after the decimal point are, in order: tenths (how many 110), hundredths (how many 1100) and thousandths (how many 11000). Example: 0.123 = 1 tenth + 2 hundredths + 3 thousandths. Read a decimal one place at a time, following the place values. Another example: 4.05 means 4 ones, 0 tenths and 5 hundredths; the 0 in the middle holds the tenths place — it has no value but cannot be skipped, or every place value shifts.
2. Block 2: Decimal AdditionThe most important rule in decimal addition is lining up the decimal points so the place values match. Example: for 3.45 + 2.1, first write 2.1 as 2.10 by padding a zero, then add in columns: 3.45 + 2.10 = 5.55. Writing 3.66 by adding the digits straight across is wrong, because the places do not line up. Another example: 0.7 + 8.36 — write 0.7 as 0.70, then 0.70 + 8.36 = 9.06; the 6 drops straight into the hundredths, and 7 + 3 = 10 in the tenths writes 0 and carries 1.
3. Block 3: Decimal SubtractionDecimal subtraction needs the same lining up, padding zeros where digits are missing. Example: for 5 - 2.34, write 5 as 5.00 and work out 5.00 - 2.34 = 2.66. Subtracting 5 - 2 straight across to get 3.34 is wrong. Padding zeros changes nothing about the size; it only helps the places line up. Another example: 7.4 - 3.75 — write 7.4 as 7.40, giving 7.40 - 3.75 = 3.65; check by adding 3.65 + 3.75 = 7.40, back to the minuend exactly.
4. Block 4: Decimal Multiplication (Basic)For decimal multiplication, multiply as whole numbers first, then count the decimal places in both factors. Example: for 2.5 × 0.4, work out 25 × 4 = 100; 2.5 has 1 decimal place and 0.4 has 1, so 2 in total, giving 1.00, which is 1. Multiply first and place the point afterwards — always in that order. Another example: 0.3 × 0.2 — work out 3 × 2 = 6, count 2 decimal places in total, and pad to get 0.06; a product smaller than both factors is normal when multiplying decimals.
5. Block 5: Decimal Division (Basic)When the divisor is a decimal, multiply both divisor and dividend by 10 or 100 to make the divisor a whole number; the quotient stays the same. Example: for 1.5 ÷ 0.3, multiply both by 10 to get 15 ÷ 3 = 5. The answer is unchanged because both numbers grew by the same factor. Another example: 7.2 ÷ 0.09 — the divisor has 2 decimal places, so multiply both by 100, giving 720 ÷ 9 = 80; once it becomes whole-number division, handle it in the usual way.
6. Block 6: Word Problem: ChangeExample: shopping costs $25.5 and you pay $50 — how much change? Work out 50.0 - 25.5 = 24.5, so the change is $24.5. With money, lining up the decimal point matters most; a whole number of dollars can be written as 50.0, and the answer should say 'the change is $24.5'. If you pay $100 for goods of $25.5, the change = 100.0 - 25.5 = $74.5; check with 74.5 + 25.5 = 100, exactly the amount paid, so the answer is sound.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Decimal add/sub: line up decimal point
• Pad zeros to align
• Decimal multiply: as whole numbers, count places
• Decimal divide: make divisor whole
• 1st digit after decimal = tenths